Where have all the cars gone? A model for determining traffic flow throughout a road network

Transcription

1 A model for determining traffic flow throughout a road network Harvey Mudd College Presentation Days 7 May 2008

2 Traffic monitoring Important Equations Lots of Traffic + Small Roads = Congestion Congestion = Bad! Two Big Questions: 1 Where are all the cars? 2 How do we find them?

3 Traffic monitoring Important Equations Lots of Traffic + Small Roads = Congestion Congestion = Bad! Two Big Questions: 1 Where are all the cars? 2 How do we find them?

4 Traffic monitoring Important Equations Lots of Traffic + Small Roads = Congestion Congestion = Bad! Two Big Questions: 1 Where are all the cars? 2 How do we find them?

5 Traffic monitoring Important Equations Lots of Traffic + Small Roads = Congestion Congestion = Bad! Two Big Questions: 1 Where are all the cars? 2 How do we find them?

6 Traffic monitoring Important Equations Lots of Traffic + Small Roads = Congestion Congestion = Bad! Two Big Questions: 1 Where are all the cars? 2 How do we find them?

7 Graph theoretic approaches A Model Represent a map as a digraph D = (V, E): Intersections The vertex set V Streets The set E of directed edges Traffic flow (cars/time) A network flow function f The flow satisfies all the normal constraints. Flow in = Flow out Sources and sinks balanced Assume knowledge of all turning ratios. Percentage of incoming flow that leaves along an arc

8 Graph theoretic approaches A Model Represent a map as a digraph D = (V, E): Intersections The vertex set V Streets The set E of directed edges Traffic flow (cars/time) A network flow function f The flow satisfies all the normal constraints. Flow in = Flow out Sources and sinks balanced Assume knowledge of all turning ratios. Percentage of incoming flow that leaves along an arc

9 Graph theoretic approaches A Model Represent a map as a digraph D = (V, E): Intersections The vertex set V Streets The set E of directed edges Traffic flow (cars/time) A network flow function f The flow satisfies all the normal constraints. Flow in = Flow out Sources and sinks balanced Assume knowledge of all turning ratios. Percentage of incoming flow that leaves along an arc

10 Example Map of Claremont

11 Example The corresponding graph

12 Definitions Definition The set of bound vertices, B, is the set of sources and sinks in the graph. The amount produced or absorbed at each bound vertex v is the balancing flow at v, denoted S v. Definition If M V, the neighbor set of M, denoted A(M), is the set of all vertices adjacent to some vertex in M.

13 Definitions Definition The set of bound vertices, B, is the set of sources and sinks in the graph. The amount produced or absorbed at each bound vertex v is the balancing flow at v, denoted S v. Definition If M V, the neighbor set of M, denoted A(M), is the set of all vertices adjacent to some vertex in M.

14 Definitions Definition The set of bound vertices, B, is the set of sources and sinks in the graph. The amount produced or absorbed at each bound vertex v is the balancing flow at v, denoted S v. Definition If M V, the neighbor set of M, denoted A(M), is the set of all vertices adjacent to some vertex in M.

15 Example a Monitored Vertex Bound Vertex Adjacent Vertex b c f d e Figure: A simple graph showing the bound vertices, a monitored vertex, and its neighbor vertices.

16 Sensor Location Problem (SLP) The Problem Given a digraph D with bound vertex set B, a flow f, and knowledge of all turning ratios, is there a set M of monitored vertices such that f can be calculated everywhere on the graph from knowledge of the flow on M? Optimization What is the smallest such set?

17 Sensor Location Problem (SLP) The Problem Given a digraph D with bound vertex set B, a flow f, and knowledge of all turning ratios, is there a set M of monitored vertices such that f can be calculated everywhere on the graph from knowledge of the flow on M? Optimization What is the smallest such set?

18 Example a Monitored Vertex Bound Vertex Adjacent Vertex S =? b b f S =? f c d S =? d 1 2 e in = 9 out = 4 S e = 5 Figure: Suppose by monitoring vertex e we observe the given flow values.

19 Example a Monitored Vertex Bound Vertex Adjacent Vertex b c f d 1 2 e in = 9 out = 4 S e = 5 Figure: Applying the turning ratios yields these edges...

20 Example a Monitored Vertex in = 11 out = 21 S = 10 b b f 5 in = 11 out = 12 S f = 1 3 c Bound Vertex Adjacent Vertex in = 12 out = 6 S d = 6 d 1 2 e in = 9 out = 4 S e = 5 Figure: And applying them once more solves the problem.

21 System of equations a : 2f ab + f ba + f ca = 0 a b f c d e

22 System of equations a : 2f ab + f ba + f ca = 0 b : f ab 3f ba + f db + f fb + S b = 0 a b f c d e

23 System of equations a : 2f ab + f ba + f ca = 0 b : f ab 3f ba + f db + f fb + S b = 0 c : f ab 2f ca + f ed = 0 d : f ba 3f db + f ed + f fb + S d = 0 e : f ca + f db 4f ed + f fb + S e = 0 f : f ba + f db + 2f ed 3f fb + S f = 0 a b f c d e

24 System of equations a : 2f ab + f ba + f ca = 0 b : f ab 3f ba + f db + f fb + S b = 0 c : f ab 2f ca + f ed = 0 d : f ba 3f db + f ed + f fb + S d = 0 e : f ca + f db 4f ed + f fb + S e = 0 f : f ba + f db + 2f ed 3f fb + S f = 0 a b f c d e

25 System of equations a : 2f ab + f ba + f ca = 0 b : f ab 3f ba + f db + f fb + S b = 0 c : f ab 2f ca + f ed = 0 d : f ba 3f db + f ed + f fb + S d = 0 f : f ba + f db + 2f ed 3f fb + S f = 0 a b f c d e

26 System of equations a : 2f ab + f ba + 3 = 0 b : f ab 3f ba S b = 0 c : f ab = 0 d : f ba S d = 0 f : f ba S f = 0 a b c f d 1 2 e

27 System of equations a : 2f ab + f ba = 3 b : f ab 3f ba + S b = 6 c : f ab = 5 d : f ba + S d = 1 f : f ba + S f = 8 a b c f d 1 2 e

28 Question When can we calculate the flow in general? When is this system of equations linearly independent?

29 Question When can we calculate the flow in general? When is this system of equations linearly independent?

30 Answer: Sometimes! Monitored Vertex Bound Vertex c Adjacent Vertex a e b d Figure: In this graph we can t calculate the flow...

31 Answer: Sometimes! Monitored Vertex Bound Vertex c Adjacent Vertex c f a e b a e b d d Figure: But in this graph, we can!

32 Failed attempts Attempt #1: A theorem was presented in [Bianco et al.(2001)] that presented a necessary and sufficient condition for calculating the flow. Attempt #2: I formulated a conjecture that modified this theorem to give a new necessary and sufficient condition for flow calculability.

33 Failed attempts Attempt #1: A theorem was presented in [Bianco et al.(2001)] that presented a necessary and sufficient condition for calculating the flow. It was wrong. Attempt #2: I formulated a conjecture that modified this theorem to give a new necessary and sufficient condition for flow calculability.

34 Failed attempts Attempt #1: A theorem was presented in [Bianco et al.(2001)] that presented a necessary and sufficient condition for calculating the flow. It was wrong. Attempt #2: I formulated a conjecture that modified this theorem to give a new necessary and sufficient condition for flow calculability.

35 Failed attempts Attempt #1: A theorem was presented in [Bianco et al.(2001)] that presented a necessary and sufficient condition for calculating the flow. It was wrong. Attempt #2: I formulated a conjecture that modified this theorem to give a new necessary and sufficient condition for flow calculability. It was wrong, too.

36 Failed attempts Attempt #1: A theorem was presented in [Bianco et al.(2001)] that presented a necessary and sufficient condition for calculating the flow. It was wrong. Attempt #2: I formulated a conjecture that modified this theorem to give a new necessary and sufficient condition for flow calculability. It was wrong, too. Well, mostly.

37 A necessary condition Theorem If there are fewer than B M vertex-disjoint paths between A(M) and B M, the flow on the graph cannot be uniquely determined by monitoring M.

38 A necessary condition Theorem If there are fewer than B M vertex-disjoint paths between A(M) and B M, the flow on the graph cannot be uniquely determined by monitoring M.

39 A necessary condition Theorem If there are fewer than B M vertex-disjoint paths between A(M) and B M, the flow on the graph cannot be uniquely determined by monitoring M.

40 A necessary condition Theorem If there are fewer than B M vertex-disjoint paths between A(M) and B M, the flow on the graph cannot be uniquely determined by monitoring M.

41 A necessary condition Theorem If there are fewer than B M vertex-disjoint paths between A(M) and B M, the flow on the graph cannot be uniquely determined by monitoring M.

42 Example Monitored Vertex Bound Vertex e Adjacent Vertex B path f b c d a Figure: B M = 2, but there is only one vertex-disjoint path from B M to A(M). Thus, we cannot calculate the flow on this graph by monitoring vertex a.

43 Example Monitored Vertex Bound Vertex Adjacent Vertex B path c a e b d Figure: In this graph, there are 2 vertex-disjoint paths from B M to A(M), but we still are unable to calculate the flow by only monitoring vertex e.

44 A necessary condition (cont.) Proof (Sketch). Menger s theorem implies #{vertex-disjoint paths} = minimum vertex cut Convert the system of equations into a matrix equation Examine a particular submatrix, and calculate the rank Apply a book-keeping argument to deduce the size of the minimum cut

45 A necessary condition (cont.) Proof (Sketch). Menger s theorem implies #{vertex-disjoint paths} = minimum vertex cut Convert the system of equations into a matrix equation Examine a particular submatrix, and calculate the rank Apply a book-keeping argument to deduce the size of the minimum cut

46 A necessary condition (cont.) Proof (Sketch). Menger s theorem implies #{vertex-disjoint paths} = minimum vertex cut Convert the system of equations into a matrix equation Examine a particular submatrix, and calculate the rank Apply a book-keeping argument to deduce the size of the minimum cut

47 A necessary condition (cont.) Proof (Sketch). Menger s theorem implies #{vertex-disjoint paths} = minimum vertex cut Convert the system of equations into a matrix equation Examine a particular submatrix, and calculate the rank Apply a book-keeping argument to deduce the size of the minimum cut

48 Computational complexity Even without a sufficient condition on the graph structure for determining flow, we can determine the complexity of SLP: Theorem SLP is NP-complete Proof (sketch). Apply a reduction from Dominating Set to the matrix formulation of the problem.

49 Computational complexity Even without a sufficient condition on the graph structure for determining flow, we can determine the complexity of SLP: Theorem SLP is NP-complete Proof (sketch). Apply a reduction from Dominating Set to the matrix formulation of the problem.

50 Computational complexity Even without a sufficient condition on the graph structure for determining flow, we can determine the complexity of SLP: Theorem SLP is NP-complete Proof (sketch). Apply a reduction from Dominating Set to the matrix formulation of the problem.

51 Future Work Determine a sufficient condition for flow calculability Find polynomial-time algorithms for specific classes of graphs (trees?) Develop approximation algorithms for the general case

52 Future Work Determine a sufficient condition for flow calculability Find polynomial-time algorithms for specific classes of graphs (trees?) Develop approximation algorithms for the general case

53 Future Work Determine a sufficient condition for flow calculability Find polynomial-time algorithms for specific classes of graphs (trees?) Develop approximation algorithms for the general case

54 Acknowledgements Prof. Susan Martonosi Dr. Kimberly Tucker Dr. Lucio Bianco Claire Connelly

55 References Lucio Bianco, Giuseppe Confessore, and Pierfrancesco Reverberi. A network based model for traffic sensor location with implications on O/D matrix estimates. Transportation Science, 35(1):50 60, Apr Lucio Bianco, Giuseppe Confessore, and Monica Gentili. Combinatorial aspects of the sensor location problem. Annals of Operations Research, 144(1): , Apr Giuseppe Confessore, Paolo Dell Olmo, and Monica Gentili. Experimental evaluation of approximation and heuristic algorithms for the dominating paths problem. Computers & Operations Research, 32(9): , Sept